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A141821 Least number k < n and coprime to n such that the largest term of the continued fraction of k/n is as small as possible. 5
1, 2, 3, 2, 5, 5, 3, 7, 3, 8, 5, 5, 11, 4, 7, 12, 13, 7, 9, 8, 17, 7, 7, 7, 19, 19, 23, 12, 11, 12, 25, 10, 13, 27, 11, 10, 9, 14, 11, 29, 11, 31, 31, 19, 17, 34, 37, 18, 19, 40, 41, 14, 17, 21, 15, 16, 17, 18, 47, 17, 23, 46, 45, 46, 25, 49, 49, 50, 29, 26, 19, 27, 31, 29, 55, 34, 61 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
See A141822 for the value of the largest term in the continued fraction of a(n)/n. Zaremba conjectured that the largest value is 5.
REFERENCES
R. K. Guy, Unsolved problems in number theory, F20.
S. K. Zaremba, ed., "Applications of number theory to numerical analysis," Proceedings of the Symposium at the Centre for Research in Mathematics, University of Montreal, Academic Press, New York, London (1972).
LINKS
Robin Visser, Table of n, a(n) for n = 2..10000 (terms n = 2..2000 from T. D. Noe).
T. W. Cusick, Zaremba's conjecture and sums of the divisor function, Math. Comp. 61 (1993), 171-176.
Takao Komatsu, On a Zaremba's conjecture for powers, Sarajevo J. Math. 1 (2005), 9-13.
EXAMPLE
For n=7, the six continued fractions for k/7 are (0, 7), (0, 3, 2), (0, 2, 3), (0, 1, 1, 3), (0, 1, 2, 2) and (0, 1, 6). It is easy to see that the fifth one, for 5/7, has the smallest maximum term, 2. Hence a(7)=5.
MATHEMATICA
Table[k=Select[Range[n-1], GCD[ #, n]==1&]; c=ContinuedFraction[k/n]; mx=Max/@c; mn=Min[mx]; k[[Position[mx, mn, 1, 1][[1, 1]]]], {n, 2, 100}]
CROSSREFS
Sequence in context: A357987 A135737 A125179 * A144308 A144307 A144310
KEYWORD
nonn
AUTHOR
T. D. Noe, Jul 08 2008
STATUS
approved

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